A small band of unconditional cooperators can make it much easier for a group to get a shared effort off the ground, but their reliability may also encourage others to slack off over the long run. A new mathematical modelling study published in Proceedings of the Royal Society B argues that steadfast minority contributors act as a double-edged sword in tasks that require a minimum number of people to succeed.
Cooperation helps both simple and complex endeavours, from bacteria forming protective films to humans funding public infrastructure. Yet giving time, energy or resources carries a personal cost, inviting free-riders who enjoy the shared benefits without contributing.
To explore when helpful behaviour endures under these pressures, the researchers used evolutionary game theory, a framework that tracks how strategies spread or fade based on payoffs. Even in anonymous, one-off encounters where helpers cannot be rewarded or punished, outcomes can shift in the presence of zealots—committed individuals who always cooperate and never switch strategy.
Earlier work suggested that a small fraction of such steady contributors could nudge a group toward cooperation. For example, a 2012 model indicated ordinary players often imitate unwavering cooperators, tipping populations toward helping behaviour when choices are not driven purely by payoff maximisation. More recently, experiments showed that committed minorities can reshape group norms, overturning established practices once they comprise about a quarter of a group.
Newer studies, however, indicate that unconditional generosity does not always boost participation, especially when individuals prioritise the highest personal return. A 2023 modelling study of spatial resource games in networked populations found that a moderate share of zealots could actually reduce overall cooperation by rewarding nearby free-riders and blocking the expansion of clusters of ordinary cooperators.
The current work, led by Yuting Dong and Lei Shi of the Yunnan University of Finance and Economics in China and Chen Shen of Kyushu University in Japan, focuses on multiplayer coordination dilemmas. These are typically modelled as stag-hunt games, where a group succeeds only if a minimum number of members contribute; failure means all lose their inputs. The team examined how a minority of zealots influences such threshold-based challenges, using purely mathematical equations and computer simulations, with code made publicly available on Zenodo.
“We wanted to understand how a minority of committed cooperators influences collective cooperation in social dilemmas,” study co-author Chen Shen explained. “Previous studies have shown that such individuals, known as ‘zealots,’ can promote cooperation in two-player social dilemma games in well-mixed populations. However, much less was known about their influence in multiplayer settings, where success requires a minimum number of individuals to cooperate. We therefore asked whether zealots are equally beneficial in these settings.”
In every round, a fixed group of players chose to cooperate by contributing one unit or to defect by contributing nothing. If the number of contributors reached the threshold, the total was multiplied and shared among all members; if not, contributors lost their input. The population included both adaptable players who could switch strategies and a minority of zealots who always contributed.
The researchers first considered an idealised, infinitely large population and tracked long‑term outcomes using replicator dynamics, a model of how strategy shares change with payoff differences. Zealots could not be distinguished from ordinary cooperators, so their presence altered group composition rather than acting through reputation or punishment.
Without zealots, there is typically a coordination barrier: if participation starts too low, cooperation collapses; if high enough, the population moves toward full cooperation or a stable mix of cooperators and free-riders. With zealots, the dynamics shift in surprising ways.
Keeping zealots to no more than half the total population, the researchers found that even a modest presence—roughly 0.2 zealots per adaptable player (about 17 percent of the population) in a 15-person game requiring at least five contributors—lowers the barrier to starting collective action. Yet under moderate returns, this same shift also lowers the stable level of cooperation in the long run, meaning fewer adaptable players cooperate once coordination is established.
“Our main finding is that zealots can have two opposing effects: they can make it easier for cooperation to emerge, but they may also discourage others from cooperating in the long run,” Shen explained. “In other words, having individuals who always cooperate does not necessarily encourage others to do the same.”
As the proportion of zealots rises, the structure of the stable outcomes changes even more. At a ratio of 0.5 zealots per adaptable player (about one‑third of the population) the coordination barrier can disappear entirely; adaptable players may reach stable coexistence under moderate returns or full cooperation under high returns, regardless of where participation began. Pushing the ratio to 0.9 zealots per adaptable player (around 47 percent of the population) can wipe out the coexistence state in parts of the moderate‑return range, leaving all adaptable players to defect.
The team also tested two extreme threshold rules. When only one contributor is needed for a reward, turning the scenario into a standard public goods game, zealots did not alter stable outcomes. Conversely, when every member must contribute for success, zealots lowered—and at higher densities eliminated—the coordination barrier, provided group payoffs exceeded individual costs.
Because real groups are finite, the researchers examined populations of limited size using a Markov chain approach, incorporating a small chance of strategy exploration. In 10‑person groups requiring at least five contributors, zealots—comprising 30 percent of the total—had little effect, with free‑riding prevailing among adaptable players. In larger, intermediate populations of 20 or 50, zealots helped push adaptables toward sustained coexistence under moderate returns. In a large population of 500, however, the same 30 percent zealot share reduced the long‑run share of adaptable cooperators compared with a zealot‑free population.
The findings explain the dual effect: a guaranteed baseline of contributions can reassure participants that the target will be met, yet once cooperation is underway that cushion allows groups to reach the threshold with fewer ordinary contributors and gives free‑riders room to coast. The authors suggest similar trade‑offs could arise in other threshold challenges, from climate action to the spread of social norms, where dedicated advocates can ease initial coordination but potentially dampen broader participation.
Shen emphasised that the results depend on factors such as the proportion of zealots, the coordination requirements, population size and the benefits of collective action. He noted potential implications for real‑world collective action challenges, including climate responses and the development of artificial intelligence, cautioning that relying on committed cooperators alone may not ensure success.
There are caveats. The work is entirely theoretical, based on mathematical equations and computer models, and omits real‑world factors such as moral norms, leadership and wealth differentials. Interactions were random and anonymous, with no opportunity for discussion or reputation effects, and the equilibrium analyses tracked only cooperation among adaptable players, not the guaranteed contributions of zealots themselves.
“Our findings should not be interpreted as suggesting that zealots are generally harmful to collective cooperation,” Shen cautioned. “Reduced cooperation among other individuals does not necessarily mean that overall cooperation declines. Moreover, our study is theoretical, and further research is needed to understand how these effects operate in real‑world settings.”
Future work could explore structured social networks, where repeated interactions occur among the same neighbours, and examine how pre‑game communication, reputation mechanisms and human or AI experiments might shape these effects. Shen described a longer‑term aim to understand how committed individuals influence participation in real‑world collective efforts, such as environmental campaigns and voluntary charitable initiatives, and to investigate how social network structures affect these dynamics through experiments involving people or AI agents.
The study, “Zealots lower coordination barriers but undermine cooperation in multiplayer games,” was authored by Yuting Dong, Hao Guo, Chen Shen, and Lei Shi.
